What can I do with an autoencoder?
An autoencoder learns to compress data, and then to decompress it again, recovering the original data.

It does this by learning a mapping from the original feature space to a lower-dimensional space, and then another mapping back. This is indeed, like PCA. The same technique is used to compress JPEG images to transmit them over the web.

Autoencoders can denoise an image because the model will not be able to easily compress or decompress random noise, and so learns to ignore it.

Autoencoders can also be used to find embeddings of data that are likely to have semantic meaning. For example, you can use an autoencoder to compress English text, and a different one to compress French text. If you add some special constraints, then the models can learn to embed in the same lower-dimensional space. This technique underlies many of the recent advances in machine translation.